By Jacobian Criterion we see that \(D\)  is a localization of a smooth sub-\(A\) -algebra \(C\subset A^{\prime }\)  containing \(B\) . Suppose that \(B=A[Y]/(f)\) , \(Y=(Y_1,\ldots ,Y_m)\) , \(f=(f_1,\ldots ,f_p)\)  and the inclusion \(v:B\rightarrow A^{\prime }\)  is given by \(Y\rightarrow y\in A^{\prime m}\) . Replacing \(B\)  by \(C\)  means in fact to substitute the system of polynomial equations \(f\)  by a system of polynomials in more variables where it is possible to apply the Implicit Function Theorem. If \(A^{\prime }\)  is Henselian then this could be very helpful. One example is the applications of Néron Desingularization to the so-called the Artin approximation property of algebraic power series rings with coefficients in an excellent Henselian DVR (see [1]}).
Actually for this aim it is not necessary the injectivity of \(v\)  and \(C\subset A^{\prime }\) . Ploski
[2]} proved that if \(A=\mathbb {C}\lbrace x\rbrace \) , \(x=(x_1,\ldots ,x_n)\) , \(B=\mathbb {C}\lbrace x,Y\rbrace /(f)\)  for some complex convergent power series \(f\)  in \(x,Y\)  then a \(\mathbb {C}\) -morphism \(v:B\rightarrow \mathbb {C}[[ x ]]\)  factors through an \(A\) -algebra of type \(\mathbb {C}\lbrace x,Z\rbrace \)  for some new variables \(Z\) .
